Theorems · Theorem · commutative algebra
Ideal.isPrime_iff_of_isPrincipalIdealRing
∀ {α : Type u} [inst : CommSemiring α] [IsPrincipalIdealRing α] {P : Ideal α},
P ≠ ⊥ → (P.IsPrime ↔ ∃ p, Prime p ∧ P = Ideal.span {p})- Defined in
- Mathlib.RingTheory.Ideal.Maximal
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 33 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommSemiringstatement and proof · cited by 10,911
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Submodule.spanproof · cited by 1,504
- Ideal.spanstatement and proof · cited by 948
- Ideal.IsPrimestatement and proof · cited by 827
- Primestatement and proof · cited by 277
- IsPrincipalIdealRingstatement and proof · cited by 131
- Prime.ne_zeroproof · cited by 47
- Ideal.span_singleton_primeproof · cited by 27
- Submodule.span_zero_singletonproof · cited by 26
Cited by1
Results whose statement or proof uses this declaration.
- Ideal.isPrime_iff_of_isPrincipalIdealRing_of_noZeroDivisorsproof · cited by 1